Past exam of the fluids and waves course of the University of Oxford / 2021 / 1 / a / ii

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- Solution ii

Solution

For a planar incompressible flow, the divergence of the velocity field must vanish. Here

\nabla \cdot u = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = - 1 + 1 = 0.

Equivalently, the Jacobian determinant of the deformation map is

det \frac{\partial ( x , y )}{\partial ( X , Y )} = e^{- t} e^{t} (cos^{2} Ω t + sin^{2} Ω t) = 1,

so every material area is preserved. However, the out-of-plane vorticity is

ω_{z} = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = - Ω (e^{2 t} + e^{- 2 t}) = - 2Ω cosh (2 t) .

Since

Ω > 0

, this vorticity never vanishes. The motion is an incompressible flow but is not an irrotational flow; its local spin is clockwise.

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